Marco went on a hiking trip. The first day he walked $24$ kilometers. Each day since, he walked a third of what he walked the day before. Which expression gives the total distance Marco walked in the first $n$ days of his trip? Choose 1 answer: Choose 1 answer: (Choice A) A $24 \cdot \dfrac{1-\left( \dfrac13 \right)^n}{1-\left( \dfrac13 \right)}$ (Choice B) B $24 \cdot \dfrac{1-\left( \dfrac23 \right)^n}{1-\left( \dfrac23 \right)}$ (Choice C) C $24 \cdot \dfrac{1-\left( \dfrac43 \right)^n}{1-\left( \dfrac43 \right)}$ (Choice D) D $24 \cdot \dfrac{1-\left( \dfrac12 \right)^n}{1-\left( \dfrac12 \right)}$
Notice that Marco's daily distances form a geometric sequence. The total distance Marco hikes after $ n$ days is the ${\text{sum}}$ of the first $n$ terms in the sequence. This is called a geometric series. This is the formula for that sum: $ S={a}\left(\dfrac{1-{r}^{ n}}{1-{r}}\right)$ where ${a}$ is the first term and ${r}$ is the common ratio. We can use this formula, along with the given information, to find the expression for the sum, $ S$. Using the given information We are given that Marco walked ${24\,\text{km}}$ on the first day. This is the first term $ a$. We are given that each daily distance is a third of the previous day's distance, or ${\dfrac13}$ times as long. This is the common ratio $ r$. We are interested in the first ${n}$ days, so the number of terms is $ {n}$. We want an expression for the total distance. This is the sum $ S$. Writing the sum $ S={24} \cdot \dfrac{1-\left({\dfrac13}\right)^{{n}}}{1-\left({\dfrac13}\right)}$ Answer The total distance Marco walked in the first $n$ days of his trip is: $24 \cdot \dfrac{1-\left( \dfrac13 \right)^n}{1-\left( \dfrac13 \right)}$